Can one use an observation level random effect in a binomial model to account for overdispersion? I had some proportion data, namely the proportion of seeds retrieved from birds. I fed different types of seeds (100 each) to a bird species and tested how different characteristics of the seeds (hardness, size) affected how many were retrieved (i.e. passed intact through the gut). With multiple replicates of course. This could have been modeled as poisson (as it was count of seed), but I decided since the seeds were bounded between 0-100 it would be better to model this as a proportion with a binomial error distribution. 
The binomial models were slightly overdispersed (phi of 3.5) so I used a observation level random effect and the overdispersion was reduced to about a factor of 1.5. 
My question is - is using these observation level random effects an appropriate way of dealing with overdispersion in binomial models. I found one forum source saying that it is a good way of doing it, but most of the materials talk about these kinds of random effects as applied to poisson models. Is there a reference which could be cited for using these random effects in binomial models?
Thanks in advance for the help!
Gail
 A: The most common cause for overdispersion are factors that influence the result but are not included in your model, either because you did not include them or were not able to observe them.
This is fairly straightforward: consider 100 observations of the number of successes of 1000 binary trials. Of these 50 obervations were taken under condition A and 50 under condition B. Under condition A the success rate is 25% and under condition B the success rate is 75%. If you ignore condition, you will get an overall success rate of 50%, but your model will be overdispersed.
Since the sample distribution of the example above is a mixture distribution of two binomial distributions, you can use this formula for the variance of a mixture distribution using the law of total variance.
$\text{Var}(X) = \text{E}(\text{Var(X}|\text{mixture component})) + \text{Var}(\text{E}(X|\text{mixture component}))$
For the case above, because the variance for a binomial distribution with $p = 0.25$ is the same as for one with $p = 0.75$ you get:
$\text{E}(\text{Var(X}|\text{mixture component})) = 1000 * 0.25 * (1 - 0.25) = 187.5$.
For the second term one can see that
$\text{Var}(\text{E}(X|\text{mixture component})) = 250^2$. The second term is much larger, so that you have a large overdispersion.
The following R code might also help:
successes.one.group <- c(rbinom(100, 1000, 0.5))

print(summary(successes.one.group))
print(var(successes.one.group))

successes.two.group <- c(rbinom(50, 1000, 0.25), rbinom(50, 1000, 0.75))

print(summary(successes.two.group))
print(var(successes.two.group))

par(mfrow = c(2, 1))
hist(successes.one.group)
hist(successes.two.group)

So, to answer your question it is clear that whenever you add a factor that is important to your data into the model (random or fixed) you should expect a reduction in overdispersion. Since you usually cannot account for or even observe everything relevant, you still need methods that can deal with overdispersion.
